scholarly journals Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series

2002 ◽  
Vol 7 (1) ◽  
pp. 41-52 ◽  
Author(s):  
A. M. López Jiménez ◽  
C. Camacho Martínez Vara de Rey ◽  
A. R. García Torres
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Direct Methods ◽  
Short Time Series ◽  
Direct Estimation ◽  
Non Linear ◽  
Short Time

The literature about non-linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation ofλstarting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non-linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.

EXTRACTING LYAPUNOV EXPONENTS FROM SHORT TIME SERIES OF LOW PRECISION

1992 ◽  
Vol 06 (02) ◽  
pp. 55-75 ◽  
Author(s):  
X. ZENG ◽  
R.A. PIELKE ◽  
R. EYKHOLT
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Polynomial Approximation ◽  
Qr Decomposition ◽  
Short Time Series ◽  
Order Polynomial ◽  
Free Parameters ◽  
Short Time ◽  
Lyapunov Exponent Spectrum

Different methods for computing the Lyapunov-exponent spectrum from a time series are reviewed. All algorithms are based on either Gram-Schmidt orthonormalization or Householder QR decomposition, and they use either the linearized map or a higher-order polynomial approximation. They also differ in implementation details. The ability to use these methods for a short time series of low precision is investigated, with special attention being given to the practicality of these algorithms; i.e., their efficiency and accuracy and the number of adjustable free parameters.

scholarly journals Constructing Digitized Chaotic Time Series with a Guaranteed Enhanced Period

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Chuanfu Wang ◽  
Qun Ding
Keyword(s):  
Time Series ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Approximate Entropy ◽  
Chaotic Time Series ◽  
Pseudorandom Sequence ◽  
Periodic Behavior ◽  
Experimental Implementation ◽  
Sequence Generator

When chaotic systems are realized in digital circuits, their chaotic behavior will degenerate into short periodic behavior. Short periodic behavior brings hidden dangers to the application of digitized chaotic systems. In this paper, an approach based on the introduction of additional parameters to counteract the short periodic behavior of digitized chaotic time series is discussed. We analyze the ways that perturbation sources are introduced in parameters and variables and prove that the period of digitized chaotic time series generated by a digitized logistic map is improved efficiently. Furthermore, experimental implementation shows that the digitized chaotic time series has great complexity, approximate entropy, and randomness, and the perturbed digitized logistic map can be used as a secure pseudorandom sequence generator for information encryption.

scholarly journals Dynamical properties of a novel one dimensional chaotic map

2022 ◽  
Vol 19 (3) ◽  
pp. 2489-2505
Author(s):  
Amit Kumar ◽  
◽  
Jehad Alzabut ◽  
Sudesh Kumari ◽  
Mamta Rani ◽  
...  
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Logistic Map ◽  
Maximal Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Dynamical Properties ◽  
Chaotic Logistic Map

<abstract><p>In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu &gt; 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.</p></abstract>

scholarly journals Non-Linear Dynamical Classification of Short Time Series of the Rössler System in High Noise Regimes

2013 ◽  
Vol 4 ◽  
Author(s):  
Claudia Lainscsek ◽  
Jonathan Weyhenmeyer ◽  
Manuel E. Hernandez ◽  
Howard Poizner ◽  
Terrence J. Sejnowski
Keyword(s):  
Time Series ◽  
Short Time Series ◽  
High Noise ◽  
Rossler System ◽  
Non Linear ◽  
Rössler System ◽  
Short Time

scholarly journals On a method of estimating chaos control parameters from time series

2018 ◽  
Vol XXI (2) ◽  
pp. 19-28
Author(s):  
Deleanu D.
Keyword(s):  
Time Series ◽  
Fixed Point ◽  
Chaos Control ◽  
Logistic Map ◽  
Control Parameters ◽  
Simple Method ◽  
Short Time Series ◽  
Measured Time ◽  
One Dimensional Maps ◽  
Short Time

The algorithm of Ott, Grebogi and Yorke (OGY) is recognized for its efficiency in controlling chaotic dynamical systems, even if the system’s equations are not known and the input data are provided by measured time series in experimental settings. Recently, Santos and Graves (SG) proposed a simple method for estimating the chaos control parameters required by OGY algorithm and applied it to the logistic map. Using only two time series of 100 values, they obtained approximate results for the fixed point case within 2 % of the analytical ones. Although the outputs refer only to a particular case, their conclusion seems to be that the method works as well as in general. To check this statement, we performed a large amount of numerical simulations on different one – dimensional maps. With slight different nuances, our findings were the same so we only presented in the paper the logistic map case. We have noticed that the use of only two short time series implies high risks in a reasonable estimate of the location of the fixed points and of the two control parameters (especially of the second). For large enough number of time series (three or five sets of 400 values each, in the paper) the results provided by numerical simulation approximated the theoretical ones within the limit of a few percent at most. The role played by each method parameter, as the radius for a close encounter of the fixed point or the number of the series and their lengths, is also investigated.

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

scholarly journals Identifying bidirectional total and non-linear information flow in functional corticomuscular coupling during a dorsiflexion task: a pilot study

2021 ◽  
Vol 18 (1) ◽  
Author(s):  
Tie Liang ◽  
Qingyu Zhang ◽  
Xiaoguang Liu ◽  
Bin Dong ◽  
Xiuling Liu ◽  
...  
Keyword(s):  
Time Series ◽  
Information Flow ◽  
Gamma Band ◽  
Motor Dysfunction ◽  
Beta Band ◽  
Short Time Series ◽  
Information Interaction ◽  
Stroke Group ◽  
Short Time ◽  
Maximal Information Coefficient

Abstract Background The key challenge to constructing functional corticomuscular coupling (FCMC) is to accurately identify the direction and strength of the information flow between scalp electroencephalography (EEG) and surface electromyography (SEMG). Traditional TE and TDMI methods have difficulty in identifying the information interaction for short time series as they tend to rely on long and stable data, so we propose a time-delayed maximal information coefficient (TDMIC) method. With this method, we aim to investigate the directional specificity of bidirectional total and nonlinear information flow on FCMC, and to explore the neural mechanisms underlying motor dysfunction in stroke patients. Methods We introduced a time-delayed parameter in the maximal information coefficient to capture the direction of information interaction between two time series. We employed the linear and non-linear system model based on short data to verify the validity of our algorithm. We then used the TDMIC method to study the characteristics of total and nonlinear information flow in FCMC during a dorsiflexion task for healthy controls and stroke patients. Results The simulation results showed that the TDMIC method can better detect the direction of information interaction compared with TE and TDMI methods. For healthy controls, the beta band (14–30 Hz) had higher information flow in FCMC than the gamma band (31–45 Hz). Furthermore, the beta-band total and nonlinear information flow in the descending direction (EEG to EMG) was significantly higher than that in the ascending direction (EMG to EEG), whereas in the gamma band the ascending direction had significantly higher information flow than the descending direction. Additionally, we found that the strong bidirectional information flow mainly acted on Cz, C3, CP3, P3 and CPz. Compared to controls, both the beta-and gamma-band bidirectional total and nonlinear information flows of the stroke group were significantly weaker. There is no significant difference in the direction of beta- and gamma-band information flow in stroke group. Conclusions The proposed method could effectively identify the information interaction between short time series. According to our experiment, the beta band mainly passes downward motor control information while the gamma band features upward sensory feedback information delivery. Our observation demonstrate that the center and contralateral sensorimotor cortex play a major role in lower limb motor control. The study further demonstrates that brain damage caused by stroke disrupts the bidirectional information interaction between cortex and effector muscles in the sensorimotor system, leading to motor dysfunction.

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